Subgroups — #14522646

Order 2000.

832 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅³	C₂	1	1	yes	maximal, characteristic
2	C₂×#92553	C₂	1	1	yes	maximal, characteristic
3	C₂×#92553	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₅³	C₂²	1	1	yes	characteristic
5	C₄×C₅³	—	2	1	no	—
6	#92553	—	2	1	no	—
7	#92553	C₄	1	2	yes	complemented
8	#87806	—	5	31	no	maximal
9	C₂×C₅³	C₂×C₄	1	1	yes	characteristic
10	C₂×C₅³	D₄	1	1	yes	characteristic
11	C₂×C₅³	Q₈	1	1	yes	characteristic
12	C₂×C₄×C₅²	D₅	1	31	yes	—
13	C₂×#1039	—	5	31	no	—
14	C₂×#1039	—	5	31	no	—
15	C₅³	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
16	C₂²×C₅²	C₂×D₅	1	31	yes	—
17	C₄×C₅²	—	2	31	no	—
18	#1039	—	5	62	no	—
19	#1039	—	10	31	no	—
20	#683	—	25	31	no	—
21	C₂×C₅²	Dic₁₀	1	31	yes	—
22	C₂×C₅²	C₄×D₅	1	31	yes	—
23	C₂×C₅²	C₂²⋊D₅	1	31	yes	—
24	C₂×C₄×C₅	C₅⋊D₅	1	31	yes	—
25	C₂×Dic₅	—	25	31	no	—
26	C₂×Dic₅	—	25	31	no	—
27	C₅²	#683	1	31	yes	complemented
28	C₂²×C₅	C₂×C₅⋊D₅	1	31	yes	—
29	C₄×C₅	—	2	31	no	—
30	Dic₅	—	25	62	no	—
31	Dic₅	—	50	31	no	—
32	C₄⋊C₄	—	125	1	no	Sylow
33	C₂×C₅	#6045	1	31	yes	—
34	C₂×C₅	C₄×C₅⋊D₅	1	31	yes	—
35	C₂×C₅	C₂²⋊(C₅⋊D₅)	1	31	yes	—
36	C₂×C₄	—	125	1	no	—
37	C₂×C₄	—	125	1	no	—
38	C₂×C₄	C₅⋊(C₅⋊D₅)	1	1	yes	characteristic
39	C₅	#87806	1	31	yes	complemented
40	C₂²	C₂×C₅⋊(C₅⋊D₅)	1	1	yes	characteristic
41	C₄	—	2	1	no	—
42	C₄	—	125	2	no	—
43	C₄	—	250	1	no	—
44	C₂	#11758769	1	1	yes	characteristic
45	C₂	C₄×C₅⋊(C₅⋊D₅)	1	1	yes	characteristic
46	C₂	C₂²⋊(C₅⋊(C₅⋊D₅))	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×C₅⋊(C₅⋊D₅)
commutator	C₂×C₅³	C₂×C₄
Fitting	C₂×C₄×C₅³	C₂
socle, prime radical, cosocle	C₂²×C₅³	C₂²
hypercenter	C₂×C₄	C₅⋊(C₅⋊D₅)
hypocenter	C₅³	C₄⋊C₄
LCS	G ▷ C₂×C₅³ ▷ C₅³
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
7. #92553	C₄	semidirect
15. C₅³	C₄⋊C₄	semidirect
27. C₅²	#683	semidirect
39. C₅	#87806	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×C₅³	2
C₂×#92553	2
#87806	5